Solve x+1/x^2-5x+6+x+5/x^2-6x+8=13/x-2

$$\frac { x + 1 } { x ^ { 2 } - 5 x + 6 } + \frac { x + 5 } { x ^ { 2 } - 6 x + 8 } = \frac { 13 } { x - 2 }$$

Solve for x

$$x=5$$

$$x = \frac{35}{11} = 3\frac{2}{11} \approx 3.181818182$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 2,3,4 since division by zero is not defined. Multiply both sides of the equation by \left(x-4\right)\left(x-3\right)\left(x-2\right), the least common multiple of x^{2}-5x+6,x^{2}-6x+8,x-2.

$$\left(x-4\right)\left(x+1\right)+\left(x-3\right)\left(x+5\right)=\left(x-4\right)\left(x-3\right)\times 13$$

Use the distributive property to multiply x-4 by x+1 and combine like terms.

$$x^{2}-3x-4+\left(x-3\right)\left(x+5\right)=\left(x-4\right)\left(x-3\right)\times 13$$

Use the distributive property to multiply x-3 by x+5 and combine like terms.

$$x^{2}-3x-4+x^{2}+2x-15=\left(x-4\right)\left(x-3\right)\times 13$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}-3x-4+2x-15=\left(x-4\right)\left(x-3\right)\times 13$$

Combine -3x and 2x to get -x.

$$2x^{2}-x-4-15=\left(x-4\right)\left(x-3\right)\times 13$$

Subtract 15 from -4 to get -19.

$$2x^{2}-x-19=\left(x-4\right)\left(x-3\right)\times 13$$

Use the distributive property to multiply x-4 by x-3 and combine like terms.

$$2x^{2}-x-19=\left(x^{2}-7x+12\right)\times 13$$

Use the distributive property to multiply x^{2}-7x+12 by 13.

$$2x^{2}-x-19=13x^{2}-91x+156$$

Subtract 13x^{2} from both sides.

$$2x^{2}-x-19-13x^{2}=-91x+156$$

Combine 2x^{2} and -13x^{2} to get -11x^{2}.

$$-11x^{2}-x-19=-91x+156$$

Add 91x to both sides.

$$-11x^{2}-x-19+91x=156$$

Combine -x and 91x to get 90x.

$$-11x^{2}+90x-19=156$$

Subtract 156 from both sides.

$$-11x^{2}+90x-19-156=0$$

Subtract 156 from -19 to get -175.

$$-11x^{2}+90x-175=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -11 for a, 90 for b, and -175 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-90±\sqrt{90^{2}-4\left(-11\right)\left(-175\right)}}{2\left(-11\right)}$$

Square 90.

$$x=\frac{-90±\sqrt{8100-4\left(-11\right)\left(-175\right)}}{2\left(-11\right)}$$

Multiply -4 times -11.

$$x=\frac{-90±\sqrt{8100+44\left(-175\right)}}{2\left(-11\right)}$$

Multiply 44 times -175.

$$x=\frac{-90±\sqrt{8100-7700}}{2\left(-11\right)}$$

Add 8100 to -7700.

$$x=\frac{-90±\sqrt{400}}{2\left(-11\right)}$$

Take the square root of 400.

$$x=\frac{-90±20}{2\left(-11\right)}$$

Multiply 2 times -11.

$$x=\frac{-90±20}{-22}$$

Now solve the equation x=\frac{-90±20}{-22} when ± is plus. Add -90 to 20.

$$x=-\frac{70}{-22}$$

Reduce the fraction \frac{-70}{-22} to lowest terms by extracting and canceling out 2.

$$x=\frac{35}{11}$$

Now solve the equation x=\frac{-90±20}{-22} when ± is minus. Subtract 20 from -90.

$$x=-\frac{110}{-22}$$

Divide -110 by -22.

$$x=5$$

The equation is now solved.

$$x=\frac{35}{11} x=5$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 2,3,4 since division by zero is not defined. Multiply both sides of the equation by \left(x-4\right)\left(x-3\right)\left(x-2\right), the least common multiple of x^{2}-5x+6,x^{2}-6x+8,x-2.

$$\left(x-4\right)\left(x+1\right)+\left(x-3\right)\left(x+5\right)=\left(x-4\right)\left(x-3\right)\times 13$$

Use the distributive property to multiply x-4 by x+1 and combine like terms.

$$x^{2}-3x-4+\left(x-3\right)\left(x+5\right)=\left(x-4\right)\left(x-3\right)\times 13$$

Use the distributive property to multiply x-3 by x+5 and combine like terms.

$$x^{2}-3x-4+x^{2}+2x-15=\left(x-4\right)\left(x-3\right)\times 13$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}-3x-4+2x-15=\left(x-4\right)\left(x-3\right)\times 13$$

Combine -3x and 2x to get -x.

$$2x^{2}-x-4-15=\left(x-4\right)\left(x-3\right)\times 13$$

Subtract 15 from -4 to get -19.

$$2x^{2}-x-19=\left(x-4\right)\left(x-3\right)\times 13$$

Use the distributive property to multiply x-4 by x-3 and combine like terms.

$$2x^{2}-x-19=\left(x^{2}-7x+12\right)\times 13$$

Use the distributive property to multiply x^{2}-7x+12 by 13.

$$2x^{2}-x-19=13x^{2}-91x+156$$

Subtract 13x^{2} from both sides.

$$2x^{2}-x-19-13x^{2}=-91x+156$$

Combine 2x^{2} and -13x^{2} to get -11x^{2}.

$$-11x^{2}-x-19=-91x+156$$

Add 91x to both sides.

$$-11x^{2}-x-19+91x=156$$

Combine -x and 91x to get 90x.

$$-11x^{2}+90x-19=156$$

Add 19 to both sides.

$$-11x^{2}+90x=156+19$$

Add 156 and 19 to get 175.

$$-11x^{2}+90x=175$$

Divide both sides by -11.

$$\frac{-11x^{2}+90x}{-11}=\frac{175}{-11}$$

Dividing by -11 undoes the multiplication by -11.

$$x^{2}+\frac{90}{-11}x=\frac{175}{-11}$$

Divide 90 by -11.

$$x^{2}-\frac{90}{11}x=\frac{175}{-11}$$

Divide 175 by -11.

$$x^{2}-\frac{90}{11}x=-\frac{175}{11}$$

Divide -\frac{90}{11}, the coefficient of the x term, by 2 to get -\frac{45}{11}. Then add the square of -\frac{45}{11} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{90}{11}x+\left(-\frac{45}{11}\right)^{2}=-\frac{175}{11}+\left(-\frac{45}{11}\right)^{2}$$

Square -\frac{45}{11} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{90}{11}x+\frac{2025}{121}=-\frac{175}{11}+\frac{2025}{121}$$

Add -\frac{175}{11} to \frac{2025}{121} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{90}{11}x+\frac{2025}{121}=\frac{100}{121}$$

Factor x^{2}-\frac{90}{11}x+\frac{2025}{121}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{45}{11}\right)^{2}=\frac{100}{121}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{45}{11}\right)^{2}}=\sqrt{\frac{100}{121}}$$

Simplify.

$$x-\frac{45}{11}=\frac{10}{11} x-\frac{45}{11}=-\frac{10}{11}$$

Add \frac{45}{11} to both sides of the equation.

$$x=5 x=\frac{35}{11}$$